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**cgiulz** Suppose that $\displaystyle f(x)$ is continuous and $\displaystyle > 0$ on $\displaystyle I = \mathbb{R},$ and $\displaystyle \displaystyle\lim_{x\to \pm\infty}f(x) = 0.$

$\displaystyle (a)$ Prove $\displaystyle f(x)$ has no minimum on $\displaystyle I.$

$\displaystyle (b)$ Prove $\displaystyle f(x)$ has a maximum on $\displaystyle I$ (note that $\displaystyle I$ is not compact).

$\displaystyle (c)$ Prove $\displaystyle (b)$ under weaker hypothesis than positivity on all of $\displaystyle I$.